On the definition of isomorphisms of linear spaces (Q1919277)

This paper contains two significant contributions towards answering the question: When is a bijection between linear spaces that maps collinear points onto collinear points an isomorphism of linear spaces? It is shown that the following conditions are sufficient for the bijection to be an isomorphism: the two spaces satisfy the exchange condition, have finite dimension, and the dimension of the range is \(\geq\) than that of the domain. Counterexamples show that the conditions on the finiteness of the dimensions involved or on the dimension inequality cannot be dropped even if the exchange condition is satisfied. The counterexample for the condition on the finiteness of dimension shows more, for it consists of a self-bijection of a linear space with the exchange axiom which maps collinear points onto collinear points, but is not an isomorphism (i.e. does not map non-collinear points to non-collinear points).